On Generalized Signal Waveforms for Satellite Navigation (797942), страница 69
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As done in previous chapters, we will generalize over n. As in (C.2), thegeneral expression for the odd case will be:BPSK ( nf c )BOC cos ( nf c 4 , f c )( f )GMod,(f )Go= Gpulse(C.29)onf c⎛⎞BOC cos ⎜⎜ f s =, f c ⎟⎟4⎝⎠We begin with n = 2 ⋅ 3 = 6 , where BOCcos(fs, fc)= BOCcos(fc, fc) can also be expressed asBCS([+1,-1,-1,+1,+1,-1], fc), being the generation matrix as follows:1{3}1{4}1{5} ⎞⎛1{0} − 1{1} − 1{2}⎟⎜1{0}1{1} − 1{2} − 1{3}1{4} ⎟⎜⎜1{0} − 1{1} − 1{2}1{3} ⎟⎟M 6 ( [+ 1,−1,−1,+1,+1,−1] ) = ⎜(C.30)1{0}1{1} − 1{2} ⎟⎜⎜1{0} − 1{1} ⎟⎟⎜⎜1{0}⎟⎠⎝In this case, the modulating term will adopt the following form,⎡⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞⎤BOCcos ( nf c 4, f c )( f ) = 6 + 2 ⎢− cos⎜⎜ 2πf ⎟⎟ − 4 cos⎜⎜ 2 2πf ⎟⎟ + cos⎜⎜ 3 2πf ⎟⎟ + 2 cos⎜⎜ 4 2πf ⎟⎟ + cos⎜⎜ 5 2πf ⎟⎟⎥GMod,o⎝ 3 fc ⎠⎝ 3 fc ⎠⎝ 3 fc ⎠⎝ 3 fc ⎠⎝ 3 f c ⎠⎦⎣(C.31)while⎛ πf ⎞⎟⎟sin 2 ⎜⎜6fBPSK (6 f c )( f ) = fc ⎝ 2 c ⎠Gpulse(πf )(C.32)318Power Spectral Density of cosine-phased BOC signalsIn the same manner, for n = 2 ⋅ 5 = 10 , BOCcos(fs, fc)= BOCcos(2fc, fc) what can also be definedin the general form as BCS([+1,-1,-1,+1,+1,-1,-1,+1,+1,-1], fc).
ThusBOCcos ( nf cGMod,o⎤⎡⎛ 2πf ⎞⎛ 2πf ⎞⎛ 2πf ⎞⎟⎟ − 8 cos⎜⎜ 2⎟⎟ + cos⎜⎜ 3⎟⎟ + ⎥⎢− cos⎜⎜⎝ 10 f c ⎠⎝ 10 f c ⎠⎝ 10 f c ⎠⎥⎢⎢⎛⎞⎛⎞⎛⎞ ⎥4, f c )( f ) = 10 + 2⎢+ 6 cos⎜⎜ 4 2πf ⎟⎟ − cos⎜⎜ 5 2πf ⎟⎟ − 4 cos⎜⎜ 6 2πf ⎟⎟ + ⎥⎢⎝ 10 f c ⎠⎝ 10 f c ⎠⎝ 10 f c ⎠ ⎥⎥⎢⎛ 2πf ⎞⎛ 2πf ⎞⎛ 2πf ⎞ ⎥⎢⎢+ cos⎜⎜ 7 10 f ⎟⎟ + 2 cos⎜⎜ 8 10 f ⎟⎟ − cos⎜⎜ 9 10 f ⎟⎟ ⎥c ⎠c ⎠c ⎠⎝⎝⎝⎦⎣(C.33)if we continue by induction we can see that the expression for any n adopts the form:⎛ πf ⎞⎟sin 2 ⎜⎜⎡n / 2nf c ⎟⎠ ⎧⎪⎡⎡ 2πf ⎤ ⎤ ⎫⎪2πf ⎤ n / 2−1ii⎝GBOCcos ( f s , fc ) = f c⎨n + 2 ⎢∑ (− 1) cos ⎢(2i − 1)⎥ + ∑ 2(− 1) (n / 2 − i ) cos ⎢2i⎥⎥⎬2nf c ⎦ i =1(πf ) ⎪⎩ ⎣ i=1⎣⎣ nf c ⎦ ⎦ ⎪⎭(C.
34)with n ∈ { 6,10,14,18,...} and n = 2 f s f c . Again, the modulating factor can be expressed as:BOC cos ( nf cGMod,o4, f c )( f ) = n + Φ 1 ( A) + Φ 2 ( A ) − Φ 3 ( A)(C.35)withΦ 1 ( A) = Φ 1+ ( A) + Φ 1− ( A)Φ 2 ( A) = Φ +2 ( A) + Φ −2 ( A)(C.36)Φ 3 ( A) = Φ 3+ ( A) + Φ 3− ( A)which is indeed the same expression we obtained in (C.7). However, since n is now twice anodd number, the results will vary slightly. Indeed it can be shown that:n/2Φ ( A) = ∑ (− 1) e+1i−A(e )2A i=e−Ai =1∑ (− e )n/22A i=e−A(− e )2 A n / 2 +1()( )n / 2 +12A+ e2A− − e2A−A e=−ee2A + 1− e2A −1i =1(C.37)since n ∈ {6,10,14,18,...} and n 2 + 1 will always be even. Thus we can simplify (C.37) asfollows:n/2( )Φ 1+ ( A) = ∑ (− 1) e − A e 2 Aiii =1=−[1 + e ] = − [1 + e ]Ane A + e−AAn(C.38)2 cosh ( A)Therefore:[][]Φ1 ( A) = ∑ (− 1) e(2i −1) A + e − (2i −1) A = ∑ (− 1) i e − A (e 2 A ) + e A (e − 2 A ) = −n/2ii =1n/2i =1ii[e+ e − An + 2e A + e− AAn](C.39)We can proceed in a similar way with Φ 2 ( A) and Φ 3 ( A) .
To do so, we will use the alreadyderived expressions for the even case and take into account that this time n ∈ { 6,10,14,18,...} .According to this,319Power Spectral Density of cosine-phased BOC signalsn / 2 −1[Φ 2 ( A) = Φ ( A) + Φ ( A) = n ∑ (− 1) e+2−2i+e2 iA− 2 iA⎡ e An − e 2 A e − An − e −2 A ⎤=n ⎢+⎥ (C.40)2A1 + e −2 A ⎦⎣ 1+ e]i =1which can be further simplified to:(eΦ 2 ( A) = − n2A) ()[+ e −2 A − e An + e − An + 2 − e A(n − 2 ) + e − A(n − 2 )](C.41)+eSimilarly, to calculate now Φ 3 ( A) we will make use of the function f ( A) defined above.(e)−A 2ANevertheless, since now n/2 is always odd, the expression simplifies as follows:(− 1)i e 2iA = ∑ (− e 2 A )i∑i =1i =1and thus, for the odd case Φ 3 ( A) is shown to be:f ( A) =Φ 3 ( A) =n / 2 −1n / 2 −1n / 2 −1[]∑ 2i(− 1) e2iA + e− 2iA =i=e An − e 2 Ae2A + 1(C.42)(n − 2)(e An + e − An ) + n [e A(n − 2 ) + e − A(n − 2 ) ] − 4(ei =1A+ e− A )2(C.43)Once we have calculated all the sum terms, it is time to obtain the expression for themodulating term of the power spectral density of the odd cosine-phased BOC modulation:⎧n +⎪− e An + e − An + 2⎪++n+⎧A−A⎪ee+⎪+ Φ ( A ) + ⎪2A−2 A4, f c )+ n e An + e − An − 2n + n e A(n − 2 ) + e − A(n − 2 )( f ) = ⎪⎨ 1= ⎨+ − n e + e+2⎪+ Φ 2 ( A ) − ⎪e A + e−A⎪⎪⎩− Φ 3 ( A)⎪ − (n − 2) e An + e − An − n e A(n − 2 ) + e − A(n − 2 ) + 4⎪+2e A + e−A⎩[BOCcos ( nf cGMod,o]() (((() [[)]])(C.44)or equivalently:(BOCcos ( nf cGMod,o4, f c )(f )=) () ((e)(4, f c )) ()⎡n e 2 A + e −2 A + 2 − e An + e − An e A + e − A − n e 2 A + e −2 A −⎤⎢⎥An− An− 2 e A + e−A + 4⎣⎢− 2n + 2 e + e⎦⎥(+eThe modulation term can also be expressed as follows:BOCcos ( nf cGMod,o)( f ) = − (eAn)(A))−A 2) ((e + e )) ()+ e − An e A + e − A + 2 e An + e − An − 2 e A + e − A + 4−A 2A(C.45)(C.46)2πfA= jnf cor2BOCcos ( nf cGMod,oAnA−− ⎞⎛ An2⎞ ⎛ A22⎜e + e ⎟ ⎜e − e 2 ⎟⎜⎟ ⎜⎟⎝⎠⎝⎠4, f c )(f )= −A−A 2e +e(2(C.47))A= j2πfnf c320Power Spectral Density of cosine-phased BOC signalsIn addition, since A=jB, we can simplify this expression as follows:2BOC cos ( nf cG Mod,o⎡⎛ B ⎞⎤⎛ Bn ⎞⎤ ⎡⎢2 cos⎜ 2 ⎟⎥ ⎢2 j sin ⎜ 2 ⎟⎥4, f c )( f ) = − ⎣ ⎝ ⎠⎦ ⎣ 2 ⎝ ⎠⎦[2 cos(B )]2⎛ πf4 cos 2 ⎜⎜⎝ fc=B=2πfnf c⎞ 2 ⎛ πf⎟⎟ sin ⎜⎜⎠⎝ nf c⎛ 2πf ⎞⎟⎟cos 2 ⎜⎜nf⎝ c ⎠⎞⎟⎟⎠ (C.48)Thus the power spectral density of BOCcos(fs=nfc/4, fc) is shown to be in the odd case:GBOCcos ( f s = nf c4, f c )BPSK ( nf c )BOCcos ( nf c( f )GMod,= Gpulseo⎛ πfcos 2 ⎜⎜⎝ fc4, f c )( f ) = 4 fc⎞ 4 ⎛ πf⎟⎟ sin ⎜⎜⎠⎝ nf c⎛⎞(πf )2 cos 2 ⎜⎜ 2πf ⎟⎟⎝ nf c ⎠⎞⎟⎟⎠ (C.49)and since n = 4 f s f c , we can also express it as follows:⎡⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎟⎟⎟ ⎥cos ⎜⎜ ⎟⎟ sin 4 ⎜⎜⎢ 2 cos⎜⎜ ⎟⎟ sin ⎜⎜4fff4f⎝ c⎠⎝ s⎠= f ⎢⎝ c⎠⎝ s ⎠⎥GBOC cos ( f s , f c ) = 4 f cc⎥⎢⎛⎞⎛πf ⎞⎟⎟(πf )2 cos 2 ⎜⎜ πf ⎟⎟πf cos⎜⎜⎥⎢⎥⎦⎢⎣⎝ 2 fs ⎠⎝ 2 fs ⎠22(C.50)As a conclusion, the normalized power spectral density of the cosine-phased BOC modulationis shown to be for n even:⎡⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎟⎟⎥sin ⎜⎜ ⎟⎟ sin 4 ⎜⎜⎢ 2 sin⎜⎜ ⎟⎟ sin ⎜⎜4 fs ⎠fc ⎠4 f s ⎟⎠ ⎥fc ⎠⎝⎝⎝⎝⎢= 4 fc= fc⎢⎥⎛ πf ⎞22 ⎛ πf ⎞⎟⎟(πf ) cos ⎜⎜ ⎟⎟πf cos⎜⎜⎢⎥⎢⎣⎝ 2 fs ⎠⎝ 2 fs ⎠⎦⎥22GBOCcos ( f s , fc )(C.51)and for n odd,⎡⎛ πf ⎞⎛ πf ⎞⎛ πf ⎞ 2 ⎛ πf ⎞ ⎤⎟⎟⎟⎥cos ⎜⎜ ⎟⎟ sin 4 ⎜⎜⎢ 2 cos⎜⎜ ⎟⎟ sin ⎜⎜4 fs ⎠fc ⎠fc ⎠4 f s ⎟⎠ ⎥⎝⎝⎝⎝⎢GBOC cos ( f s , f c ) = 4 f c= fc⎥⎢⎛ πf ⎞22 ⎛ πf ⎞⎟⎟(πf ) cos ⎜⎜ ⎟⎟πf cos⎜⎜⎥⎢⎥⎦⎢⎣⎝ 2 fs ⎠⎝ 2 fs ⎠22(C.52)Now that we have derived the expressions of the power spectral density of the sine andcosine-phased BOC modulations, it is interesting to note the following relationship:⎛ ⎛ πf ⎞ ⎛ πf⎜ sin ⎜⎟ sin ⎜⎜ ⎜⎝ 2 f s ⎟⎠ ⎜⎝ f cGBOC sin ( f s , f c ) ( f ) = f c ⎜⎛⎞⎜ πf cos⎜ πf ⎟⎜2f ⎟⎜⎝ s⎠⎝22⎛⎞⎞⎛⎞ ⎛ ⎞ ⎞⎜ 2 sin 2 ⎜ πf ⎟ sin ⎜ πf ⎟ ⎟⎟⎟ ⎟⎜⎟ ⎜ ⎟⎠⎟ = f ⎜⎝ 4 f s ⎠ ⎝ f c ⎠ ⎟ = GBOC cos ( f s , f c ) ( f )c⎜⎟⎛⎞ ⎛⎞⎟⎛ πf ⎞⎟⎜ πf cos⎜ πf ⎟ tan⎜ πf ⎟ ⎟⎟⎟tan 2 ⎜⎜⎜2f ⎟ ⎜4f ⎟⎟⎟⎜⎝ s ⎠ ⎝ s ⎠⎠⎝ 4 fs ⎠⎠⎝(C.53)321Power Spectral Density of cosine-phased BOC signalsthat allows us to go from the sine-phased expression to the other one.
We show in the nextfigure the sine-phased BOC modulation together with its cosine-phased counterpart and theinverse tangent term of the expression above that relates both. For simplicity a sub-carrierfrequency fs of 1.023 MHz and a carrier frequency fc of 1.023 MHz were assumed.Figure C.1. Power Spectral Density of Sine-phased, Cosine-phased and Inverse TangentFunction of BOC(10,5)322Power Spectral Density of TCS signalsDAppendix.
PSD of TCS SignalsThe TCS signal waveform has been introduced in chapter 4.5.1. In the next lines we willderive the power spectral density of an arbitrary TCS(fs, fc,ρ) with ρ allocated symmetricallyaround the borders of the sub-chip. To facilitate the understanding, refer to Figure 4.25 inchapter 4.5.1.For the most general case, the Fourier transform of the signal is shown to be:n( k − ρ / 2 )Tck =1n( k −1+ ρ / 2 )TcnS TCS ( jω ) = ∑ ∫sk e− jωt⎡ ωT⎤dt = sin ⎢ c (1 − ρ )⎥ eω ⎣ 2n⎦2jωTc2nn∑sk =1ke−jkωTcn(D.1)or in the frequency domainjπfn−⎡ πf⎤1STCS ( f ) = sin ⎢ (1 − ρ )⎥ e nf c ∑ sk eπfk =1⎣ nf c⎦j 2 kπfnf c(D.2)Thus, the general expression for the power spectral density of the TCS adopts the followingform:⎡ πf⎤sin 2 ⎢ (1 − ρ )⎥ n2πfk 2−jnf⎛ fc ⎞BCS([s ], f c )TCSc⎣⎦( f ) = ⎜⎜GTCS([ s ], f c ) ( f ) = Gpulse ( f )GModsk e nf c(D.3)⎟⎟∑2(πf )k =1⎝1− ρ ⎠As we can recognize from (D.3), the modulating term on the right is common to the MCSdefinition and therefore all the results that we obtained in the previous Appendixes apply heretoo.
This will allow us derive general expression for all possible cases of TCS signals. Indeed,the PSD of the TCS modulation can also be expressed as:⎡ πf⎤sin 2 ⎢ (1 − ρ )⎥ nn −1 n⎛ f ⎞ωT ⎤ ⎫⎡⎣ nf c⎦ ⎧ s2 + 2GTCS([ s ], f c ) ( f ) = ⎜⎜ c ⎟⎟si s j cos ⎢( j − i ) c ⎥ ⎬⎨∑ i∑∑2n ⎦⎭(πf )⎣i =1 j = i +1⎝1− ρ ⎠⎩ i =1(D.4)Since TOC([s], fc) is a particular case of TCS, we can distinguish the following cases:•Even TOCsin([s], fc): In this case the modulating term coincides with that ofBOCsin(fs , fc) for the even case, which was already shown in (B.11) and thus:⎡ ⎡ πf⎡ πf⎤⎛ πf ⎞⎤ ⎛ πfsin ⎢ (1 − ρ )⎥ sin 2 ⎜⎜ ⎟⎟⎢ sin ⎢ (1 − ρ )⎥ sin ⎜⎜⎛ f ⎞⎣ nf c⎦⎝ f c ⎠ = ⎛⎜ f c ⎞⎟ ⎢ ⎣ nf c⎦ ⎝ fcGTCS([ s ], f c ) ( f ) = ⎜⎜ c ⎟⎟2⎜⎟⎛ πf ⎞ ⎝ 1 − ρ ⎠ ⎢⎛ πf ⎞(πf )⎝1− ρ ⎠⎟⎟cos 2 ⎜⎜πf cos⎜⎜ ⎟⎟⎢⎢⎣⎝ nf c ⎠⎝ nf c ⎠2•⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(D.5)Odd TOCsin([s], fc): In this case the modulating term coincides with that ofBOCsin(fs , fc) for the odd case, which was already shown in (B.23).
Thus:323Power Spectral Density of TCS signals⎡ ⎡ πf⎡ πf⎤⎛ πf ⎞⎤ ⎛ πfsin ⎢ (1 − ρ )⎥ cos 2 ⎜⎜ ⎟⎟⎢ sin ⎢ (1 − ρ )⎥ cos⎜⎜⎛ f ⎞⎣ nf c⎦⎝ f c ⎠ = ⎛⎜ f c ⎞⎟ ⎢ ⎣ nf c⎦ ⎝ fcGTCS([ s ], f c ) ( f ) = ⎜⎜ c ⎟⎟2⎜⎟⎛ πf ⎞ ⎝ 1 − ρ ⎠ ⎢⎛ πf ⎞(πf )⎝1− ρ ⎠⎟⎟cos 2 ⎜⎜πf cos⎜⎜ ⎟⎟⎢⎢⎣⎝ nf c ⎠⎝ nf c ⎠2••⎞⎤⎟⎟ ⎥⎠⎥⎥⎥⎥⎦2(D.6)Even TOCcos([s], fc): In this case the modulating term coincides with that ofBOCcos(fs , fc) for n even. This was already shown in (C.26).
Accordingly,⎡ πf⎤⎛ πf ⎞⎛ πf ⎞⎟⎟sin 2 ⎢ (1 − ρ )⎥ 4 sin 2 ⎜⎜ ⎟⎟ sin 2 ⎜⎜nffnf⎛ fc ⎞⎣ c⎦⎝ c⎠⎝ c⎠=GTCS([ s ], f c ) ( f ) = ⎜⎜⎟⎟2−ρ1⎛⎞()fπfπ2⎝⎠⎟⎟cos 2 ⎜⎜⎝ nf c ⎠(D.7)2⎡⎡ πf⎤ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎥⎢ 2 sin ⎢ (1 − ρ )⎥ sin ⎜⎜ ⎟⎟ sin ⎜⎜nf cf c ⎠ ⎝ nf c ⎟⎠ ⎥⎛ fc ⎞ ⎢⎣⎦⎝= ⎜⎜⎟⎟⎥⎛ πf ⎞⎝1− ρ ⎠ ⎢πf cos⎜⎜ ⎟⎟⎥⎢⎥⎦⎝ nf c ⎠⎣⎢Odd TOCcos([s], fc): In this case the modulating term coincides with that ofBOCcos(fs , fc) for n odd which was already shown in (C.48). As a result:⎡ πf⎤⎛ πf ⎞⎛ πf ⎞⎟⎟sin 2 ⎢ (1 − ρ )⎥ 4 cos 2 ⎜⎜ ⎟⎟ sin 2 ⎜⎜nffnf⎛ fc ⎞⎣ c⎦⎝ c⎠⎝ c⎠=GTCS([ s ], f c ) ( f ) = ⎜⎜2⎟⎟−ρ1⎛⎞()fπfπ2⎝⎠⎟⎟cos 2 ⎜⎜⎝ nf c ⎠(D.8)2⎡⎡ πf⎤ ⎛ πf ⎞ ⎛ πf ⎞ ⎤⎟⎥⎢ 2 sin ⎢ (1 − ρ )⎥ cos⎜⎜ ⎟⎟ sin ⎜⎜nf cf c ⎠ ⎝ nf c ⎟⎠ ⎥⎛ fc ⎞ ⎢⎣⎦⎝= ⎜⎜⎟⎟⎥⎛ πf ⎞⎝1− ρ ⎠ ⎢πf cos⎜⎜ ⎟⎟⎥⎢⎥⎦⎢⎣⎝ nf c ⎠which coincides with the well known expressions that can be found in the literature.324Power Spectral Density of UTCS signalsEAppendix.
PSD of UTCS SignalsA sub-carrier waveform of great interest to model a generic signal is studied in this Appendix.In the next lines we will derive the power spectral density of an arbitrary UnilateralTCS(fs, fc, ρ) or UTCS for short. Unlike the TCS case that we studied in Appendix D, here thezero support is only on one of the sides of the sub-chips and not symmetrically placed.Figure E.1.
Chip waveform of the UTCS modulationAccording to the figure above, the Fourier transform of the signal is shown to be:nkTcS UTCS ( jω ) = ∑ ∫ (kn− ρ )Tc sk e − jωt dt =k =1n⎡ ωT ⎤sin ⎢ ρ c ⎥ eω ⎣ 2n ⎦2jωρTc2nn∑ sk ek =1−jkωTcn(E.1)and in the frequency domainjπfρ⎡ πf ⎤ nf c1S UTCS ( f ) = sin ⎢ ρ⎥eπf⎣ nf c ⎦n∑sk =1ke−j 2 kπfnf c(E.2)Thus, the general expression for the power spectral density of a generic UTCS signal will be:⎡ πf ⎤sin 2 ⎢ ρ2πfk 2⎥ n−jnf⎛ fc ⎞c⎦⎣GUTCS([ s ], f c ) ( f ) = ⎜⎜sk e nf c(E.3)⎟⎟∑2k =1⎝ 1 − ρ ⎠ (πf )It must be noted that in the definition of the unilateral tertiary coded symbols, the parameter ρrepresents the period of time with no zero dwell as was the case of the TCS. Again, we canrecognize from (E.3) that the Power Spectral Density can be separated in the two usual terms,namely the pulse term and the modulation term:BCS([s ], f c )UTCS( f )GMod(f )GUTCS([ s ], f c ) ( f ) = Gpulse(E.4)This expression is very convenient, since all the modulating terms derived in previousAppendixes can also be applied here.325Power Spectral Density of UTCS signals326Power Spectral Density of 8-PSK sine-phased BOC signalsFAppendix.
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